Artificial Tribotactic Microscopic Walkers: Walking Based
on Friction Gradients
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Steimel, Joshua P., Juan L. Aragones, and Alfredo, AlexanderKatz. "Artificial Tribotactic Microscopic Walkers: Walking Based
on Friction Gradients." Phys. Rev. Lett. 113, 178101 (October
2014). © 2014 American Physical Society
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http://dx.doi.org/10.1103/PhysRevLett.113.178101
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American Physical Society
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PHYSICAL REVIEW LETTERS
PRL 113, 178101 (2014)
Artificial Tribotactic Microscopic Walkers: Walking Based on Friction Gradients
Joshua P. Steimel, Juan L. Aragones, and Alfredo Alexander-Katz*
Department of Materials Science and Engineering, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA
(Received 10 April 2014; published 22 October 2014)
Friction, the resistive force between two surfaces sliding past each other, is at the core of a wide diversity
of locomotion schemes. While such schemes are well described for homogeneous environments,
locomotion based on friction in inhomogeneous environments has not received much attention. Here
we introduce and demonstrate the concept of tribotaxis, a motion that is guided by gradients in the friction
coefficient. Our system is composed of microwalkers that undergo an effective frictional interaction with
biological receptors on the substrate, which is regulated by the density of such receptors. When actuated
stochastically, microwalkers migrate to regions of higher friction, much like a chemotactic cell migrates to
regions of higher chemoattractant concentration. Simulations and theory based on biased random walks are
in excellent agreement with experiments. We foresee important implications for tribotaxis in artificial and
natural locomotion in biological environments.
DOI: 10.1103/PhysRevLett.113.178101
PACS numbers: 87.85.gj, 81.40.Pq, 87.16.Uv
Tribotaxis is the process by which an active object, biotic
or abiotic, detects differences in the effective local friction
coefficient and moves to regions of higher or lower friction
according to a given protocol. The local friction coefficient
between an object and a surface is dictated by the effective
interactions between both. If these interactions are directional in nature, the friction coefficient is anisotropic.
A prominent example of this, being the skin of many
animals which feels rough when stroked in one direction,
yet soft in the other. The origin of this asymmetry is due
to the directionality and ordering of hair or scales sticking
out from the surface at a slanted angle, which helps in
modulating the effective friction between the skin (or scales)
of an animal and the surrounding fluid [1,2]. These types
of materials are important in many other processes, such as
regulating the flow of complex fluids [3], controlling the
motion of cells [4], or even skiing up a mountain.
While the motifs that give rise to these asymmetric
friction coefficients are rather large, one can envision an
alternative microscopic scenario. For example, one can
think of exchanging the mechanical texture with a chemical
texture in which friction is dominated by the strength and
spatial density of reversible bonds between an object and a
substrate [5,6]. Moreover, gradients in the spatial density of
such ligands produce anisotropic friction coefficients. Live
cells, for example, naturally detect surface ligand gradients
and move accordingly [7]; this is one of the most important
clues for locomotion since cells are constantly encountering
surfaces in our bodies. Mimicking this behavior using
biological ligand-receptor pairs can potentially allow one
to “walk on” and sense different conditions in the vast
amounts of interfaces in tissues and organs. In addition,
chemically based tribotaxis can be used to locally sense
friction in purely synthetic environments. Interestingly,
despite the tremendous experimental and theoretical effort
0031-9007=14=113(17)=178101(5)
in the last years in designing artificial active systems that
can perform chemotaxis in an intrinsic way [8–10], the
design of a chemotactic artificial active system that detects
biological molecules remains a challenge.
Here, we have developed a tribotactic system able to detect
gradients in friction due to gradients in the density of
biological receptors, thus effectively performing chemotaxis
[11–13]. The system is a magnetically assembled colloidal
doublet (a microwalker), which converts rotational motion
into translational motion due to enhanced friction near a
surface. Clearly, for weak friction scenarios the doublet
primarily slips and translates slowly while for higher friction
coefficients the doublet will walk faster in a hinge-type
fashion [see Fig. 1(a)]. The effective friction for this system
can be tuned by functionalizing the surface of the magnetic
beads and the substrate with complementary ligand-receptor
pairs [Fig. 1(b)]. Thus, by controlling the spatial density
of ligands on the surface it is possible to create gradients in
the friction coefficient. From nonequilibrium mechanics, it
is known that if a particle is stochastically driven on such
gradients, its motion resembles that of biased random walks
[14]. To create such stochastic walks, we utilize a particular
protocol that can be best described in a two-part scheme
per random step. In the first part, the so-called move step,
we rotate the microwalker by applying a torque in a given
direction for a fixed time τ. In the second part, the so-called
relax step, we allow the doublet to lie flat for a period of
time τrelax , which increases the probability of forming bonds
with the substrate. We repeat this protocol for the total
number of steps of the walk and randomize the direction of
the torque per step according to a one-dimensional random
walk. Clearly, over the course of multiple steps following
the protocol previously described, the microwalker will drift
toward regions of higher ligand density where friction is
higher, eventually becoming trapped in such regions.
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FIG. 1 (color online). Schematic representation of the tribotactic system. (a) A superparamagnetic doublet, or microwalker,
spontaneously self-assembles upon actuation of the magnetic field, B. To rotate the magnetic doublet on the substrate we couple the
doublet to a rotating magnetic field. The microwalker is actuated for a time period τ and is then allowed to relax for a certain period τrelax ;
this constitutes one random walk step. The motion of the center of mass of the tribotactic walkers is, in general, denoted by δ.
In particular, the walkers tend to perform larger displacements δ2 when moving toward regions of higher friction and display smaller
steps toward regions of lower friction δ1 . Hence, the microwalker will drift toward regions of higher friction, effectively migrating
toward areas with a higher ligand density. (b) The effective friction (or interaction) between the beads and the substrate is controlled by
functionalizing both surfaces with complementary ligand-receptor pairs. In this Letter the beads are functionalized with streptavidin and
the substrate with biotin. The beads, 3 μm in diameter, are further functionalized with biotinylated polyethylene glycol (mPEG-biotin)
to mask and reduce the effective binding strength between the walker and the substrate. Free avidin is also used to block available
binding sites on the substrate.
As mentioned above, our tribotactic system is composed
of two superparamagnetic beads coated with streptavidin
and a biotinylated substrate. To modulate the surface and
reduce the binding strength of the biotin-streptavidin bond,
we decorate such beads with biotinylated PEG, leaving
only a few active sites on the surface. The density and
distribution of available biotin sites on the substrate can
also be screened by using free avidin [see Fig. 1(b)]. The
gradients in friction were created by drying a water droplet
(50 μl) containing free avidin ligands at a concentration of
1 mg=ml and left to evaporate on a biotinylated substrate
[see Fig. 2(a)]. The capillarity flows induced by the
differential evaporation rates across the droplet carry most
of the avidin ligands to the edge, thereby producing a socalled coffee-ring pattern along the perimeter of the droplet
[15]. Interestingly, a robust annular friction pattern develops around the center of the droplet [see Fig. 2(b)]. Notice
that such patterns have also been observed in other
evaporating systems [16]. The resulting frictional landscape
exhibits areas of high friction separated by low friction
“valleys.” This can be seen in Fig. 2(c), where we plot
the velocity of translation of the doublet vs the distance to
the center of the droplet. In this case we continuously allow
the microwalker to move, effectively setting τ → ∞. Note
that higher velocities correspond to higher effective friction. In particular, we find the velocity at x ≈ 0 μm and
x ≈ 800 μm to be approximately 30–40% larger than at the
lowest point, corresponding to x ¼ 450 μm. The velocity
displayed in this latter region corresponded to a biotin
substrate that was completely coated with avidin ligands
(the horizontal red dashed line), which was confirmed by an
independent measurement on a surface with no free biotin
(Fig. S4a [17]). Assuming a simple Stokes drag scenario,
the variations in velocity imply a difference in effective
friction forces between both regions of less than 60 fN.
Thus, our microwalkers are extremely sensitive to minute
variations in the friction coefficient. For more details about
the system, the materials, and the methods we refer the
reader to the Supplemental Material [17].
The tribotactic nature of our microwalkers was probed
by actuating them with a rotating magnetic field where the
direction of rotation is varied in a stochastic fashion, as in a
one-dimensional random walk. The total number of steps in
each walk is set to 300, and we allow the system to relax for
τrelax ¼ 10 s in between steps, which provides enough time
for the doublet to lie parallel to the substrate. The period of
actuation, τ, was set at 2 s. In Fig. 3(a), the trajectories
for several independent random walks originating from
different points in the sample are plotted. As is clear from
this plot, the doublets initially positioned at x ¼ 0 and
x ¼ 800 μm become trapped in these regions with a larger
friction coefficient. Walkers initially positioned at regions
with a lower coefficient of friction, like x ¼ 300 and
x ¼ 600 μm, drift along the gradient in the direction of
stronger friction. The region with the lowest friction is at
an unstable point and the microwalkers can go either
way. This is clearly observed for the walkers initially
positioned around x ¼ 450 μm, which drifted in both
directions [see green trajectories in Fig. 3(a)]. For microwalkers starting at x ¼ 300 μm and x ¼ 600 μm, we
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(a)
(b)
(c)
FIG. 2 (color online). Creating gradients in the density of biotin
binding sites and mapping the velocity profile of the substrate.
(a) A water droplet containing avidin is placed on the biotinylated
substrate and left to evaporate for 24 hrs. (b) Schematic of the
gradient in the density of available biotin binding sites based on
the frictional landscape obtained by measuring the local walking
velocities shown below. (c) Average velocity profile (μm=s) of the
microwalkers as a function of the distance to the center of the
droplet in μm. The origin (x ¼ 0 μm) corresponds to the center of
the avidin droplet. The horizontal red line denotes the velocity of
the walkers on a fully passivated biotin surface, measured in an
independent experiment. The colored dots denote the initial
positions from which the random walks in Fig. 3 were launched.
The frequency of rotation is set to 1 Hz.
computed the drift velocity from the slope of a linear fit
to their trajectories [see Figs. 3(b) and 3(c)]. The drift
was estimated to be u ¼ 0.32 μm=s (x ¼ 300 μm) and
u ¼ 0.28 μm=s (x ¼ 600 μm). Notice, however, that the
drift velocity of these tribotactic microwalkers can be tuned
by modifying the parameters of the system, such as the
actuation protocol or the size of the walkers. Furthermore,
our approach is extremely versatile because the beads can
be functionalized with a multitude of biologically relevant
motifs or can even be multiplexed, meaning they can have
combinations of motifs to explore cooperative effects or
multiple gradients. Thus, such tribotactic microwalkers can
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be used as sensitive chemotactic probes in biological
environments. Notice that the origin of the tribotactic
response in our system is due to the frictional landscape
of the environment and not on the presence of a (nonbiological) fuel source [10,18,20], and thus the chemotactic
landscape is static. Hence, our tribotactic walkers complement current approaches to create artificial chemotactic
systems.
To further understand our tribotactic system, we analyzed the distances traveled by the walkers at each step for
both directions. A probability distribution of such displacements is shown in the insets of Figs. 3(b) and 3(c) for
the doublets positioned at x ¼ 600 μm and x ¼ 300 μm,
respectively. From these distributions it is clear that the
tribotactic walkers exhibit, on average, asymmetric displacements depending on the direction of motion. The
average displacement toward regions with weaker friction
is denoted by δ1 ðxÞ and toward regions with stronger
friction is denoted as δ2 ðxÞ, and their corresponding
standard deviations are σ 1 ðxÞ and σ 2 ðxÞ (see insets in
Fig. 3). It is well known that such asymmetric distributions
lead to directed motion (drift). In order to see this, one can
develop a master-equation-based model of the stochastic
motion exhibited by the doublets that incorporates such
distributions of microscopic displacements. By deriving
a Fokker-Planck equation via a truncated Kramers-Moyal
expansion of the original master equation [21], it is
possible to solve the time evolution of the probability
distribution of the position of the walkers. This model
equation has previously been used to describe chemotaxis
[22–26]. Note that the actuation protocol is that of a random
walk and thus has no memory. The model and subsequent
derivation of the governing equations can be found in
the Supplemental Material [17]. The final equation that
describes the temporal evolution of the probability density
is given by
∂Pðx; tÞ
∂
1 ∂2
¼ − ½uðxÞPðx; tÞ þ
½DðxÞPðx; tÞ;
∂t
∂x
2 ∂x2
ð1Þ
where uðxÞ and DðxÞ are the drift velocity and diffusion
coefficient of the walkers, respectively. These coefficients
have a microscopic origin, and their functional form can be
written in terms of the displacements δ1 ðxÞ and δ2 ðxÞ as
uðxÞ ¼ (δ2 ðxÞ − δ1 ðxÞ)=τ and DðxÞ ¼ (δ1 ðxÞ2 þ δ2 ðxÞ2 )þ
(σ 1 ðxÞ2 þ σ 2 ðxÞ2 )=τ. The first term is responsible for
tribotactic motion and represents the drift due to the
asymmetry in displacements, while the second term corresponds to diffusive motion. Assuming the magnitude of
the displacements is independent of the position (i.e.,
u ¼ constant), which is the case we find experimentally
(see Fig. S6b [17]), the solution to Eq. (1) is a normal
distribution whose average drifts with time at a velocity u,
and the width of the distribution grows as t1=2 . Such growth
can be better seen from simulated random walks with
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FIG. 3 (color online). Trajectories of tribotactic microwalkers driven stochastically. (a) Random-walk trajectories for doublets initially
placed at approximately 0, 300, 450, 600, and 800 μm from the origin. The color of the walk corresponds to the highlighted position in
Fig. 2(c). A drift toward regions with a higher friction coefficient is observed. Walkers initially placed in regions with the maximum
friction coefficient became trapped. (b) Detailed trajectories for walkers initially placed at 600 μm. The solid, dashed, and dotted lines
correspond to a linear fit to the experimental trajectories (slope 0.28 μm=s), the simulated trajectories, and the theoretical drift velocity
(u ¼ ½δ2 ðxÞ − δ1 ðxÞ=τ), respectively. (Inset) Probability distribution of the displacement in each direction. δ1 ðxÞ corresponds to the
mean displacement in the direction of lower friction, while δ2 ðxÞ denotes the average displacement toward higher friction areas.
(c) Trajectories for walkers initially placed at 300 μm. δ1 ðxÞ and δ2 ðxÞ are defined as in part B. The different lines correspond, as in part
B, to the linear fits to the experimental (solid line), simulated (dashed line), and theoretical (dotted line) results. The slope of the linear fit
to the experimental data is 0.32 μm=s.
displacements distributed according to those found experimentally (see Fig. S5 [17]). The drift velocities from such
simulated experiments are in excellent agreement with the
theoretical and experimental values, as can be seen by
comparing the dashed (simulated trajectories), dotted
(analytical model), and solid (experimental trajectories)
lines in Figs. 3(b) and 3(c).
As described above, the regions of highest friction act
as tribotactic traps. To measure the relative strength of
these traps with respect to the spatial step size of the
random walks, we performed longer stochastic sequences
of 500 steps starting at the center of the trap. The effective
spatial step size was varied by changing the actuation
period, τ, of the rotating magnetic field from 2 to 20 s. The
resulting trajectories were analyzed, and the corresponding probability distributions are presented in Figs. 4(a)
and S7a [17]. The solid lines in Fig. 4(a) represent the
numerical solutions based on Eq. (1) of our tribotactic
model, including the directional dependence of the drift
term due to the trap [17]. We also simulated many more
repetitions of such random walks to get better statistics
[17]. As expected, the increase in τ translates into a
broadening of the steady-state distributions. In light of the
Gaussian-like shape of the steady-state distribution, we
evaluated the stiffness of the trap κ using the standard
deviation, σ, of such distribution from Fig. 4(a). From
statistical mechanics we know that the probability distribution of an harmonic oscillator can be written as
PðxÞ ∝ exp ð− 12 κðτÞx2 Þ, where κ ¼ 1=σ 2 . In Fig. 4(b)
we can see that κ ∝ 1=τ2 ; thus, τ plays the role of a
square root of a temperature by analogy to the simple
harmonic well. To corroborate the results from the
steady-state distributions, we calculated also the rootmean-square displacement (RMSD) for the trajectories.
As expected, the RMSD, for both simulations and
experiments, exhibits a plateau at long times for all of
the different τ. Such behavior is characteristic of a
confined random walker and directly corroborates the
presence of a tribotactic trap.
FIG. 4 (color online). Tribotactic trap. (a) Experimental probability distribution of position of tribotactic walkers with an
actuation period, τ, of 2 s (purple), 5 s (blue), 10 s (red) and 20 s
(green), respectively. Solid lines correspond to the numerical
solutions to Eq. (1) using the experimental parameters for the step
displacement distribution. (b) Stiffness of the tribotactic trap, κ,
computed from the standard deviation of the distributions
obtained experimentally (squares), using simulations (circles),
and using the analytical model (diamonds). The dashed line
represents a fit to the data assuming the functional form κ ∝ τ−2 .
(c) RMSD calculated from the experimental (darker colors) and
simulation (lighter colors) trajectories for actuation periods of 2 s
(purple), 5 s (blue), 10 s (red) and 20 s (green). The RMSDs show
the characteristic behavior of a confined random walk.
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In summary, we have demonstrated tribotaxis using a
superparamagnetic doublet functionalized with biological
ligands. Such a microwalker exploits friction to sense
ligand concentration gradients and migrates toward the
higher density regions, eventually becoming trapped in
such regions. To avoid trapping, we have shown that
increasing the actuation time yields a “softening” of the
trap and can in principle lead to escape, similar in spirit to
the way bacteria escape from their own chemoattractant
[27]. Thus, the frictional landscapes felt by these tribotactic
microwalkers can be tuned, making this system ideal to
measure differences in friction coefficients in complex
environments or to find areas with relatively higher density
of ligands without a priori knowledge. Furthermore, given
the similarities between cells and the aforementioned
microwalkers in terms of size and the environment in
which they move, we believe that nature may already be
using friction as an ultrasensitive probe for differences in
the density of ligands which ultimately play a determining
role in biological processes.
We are grateful for the generous support of the Chang
Family (J. P. S. and A. A.-K.), the MITEI BP Fellowship
Program (J. P. S.), and the Department of Energy BES
Award No. ER46919 (J. L. A. and A. A.-K.). We thank
David Bono and Stephanie Moran for their help in
developing the first experimental setups. We also thank
Diego Lopez and Michael Alemayhu who help us with the
particle tracking during the early stages of this project.
*
aalexand@mit.edu
[1] G. D. Bixler and B. Bhushan, Soft Matter 8, 11271 (2012).
[2] J. Hazel, M. Stone, M. S. Grace, and V. V. Tsukruk,
J. Biomech. 32, 477 (1999).
[3] D. Quéré, Annu. Rev. Mater. Res. 38, 71 (2008).
[4] M. Le Berre, Y.-J. Liu, J. Hu, P. Maiuri, O. Bénichou,
R. Voituriez, Y. Chen, and M. Piel, Phys. Rev. Lett. 111,
198101 (2013).
[5] J. Klein, Annu. Rev. Mater. Sci. 26, 581 (1996).
[6] B. Bhushan, J. N. Israelachvili, and U. Landman, Nature
(London) 374, 607 (1995).
[7] S. B. Carter, Nature (London) 213, 256 (1967).
week ending
24 OCTOBER 2014
[8] S. Granick, S. K. Kumar, E. J. Amis, M. Antonietti,
A. C. Balazs, A. K. Chakraborty, G. S. Grest, C. Hawker,
P. Janmey, E. J. Kramer, R. Nuzzo, T. P. Russell, and
C. R. Safinya, J. Polym. Sci., Part B: Polym. Phys. 41,
2755 (2003).
[9] A. Bhattacharya, O. B. Usta, V. V. Yashin, and A. C. Balazs,
Langmuir 25, 9644 (2009).
[10] S. J. Ebbens and J. R. Howse, Soft Matter 6, 726 (2010).
[11] H. C. Berg and D. A. Brown, Nature (London) 239, 500
(1972).
[12] J. Alder, Science 153, 708 (1966).
[13] H. C. Berg, Annu. Rev. Biophys. Bioeng. 4, 119
(1975).
[14] E. A. Codling, M. J. Plank, and S. Benhamou, J. R. Soc.
Interface 5, 813 (2008).
[15] L. Shmuylovich, A. Q. Shen, and H. A. Stone, Langmuir 18,
3441 (2002).
[16] D. Kaya, V. A. Belyi, and M. Muthukumar, J. Chem. Phys.
133, 114905 (2010).
[17] See Supplemental Material at http://link.aps.org/
supplemental/10.1103/PhysRevLett.113.178101, which includes Refs. [14–16,18,19,22], for a more detailed description of the experimental protocol, data analysis and the
analytical model.
[18] W. F. Paxton, K. C. Kistler, C. C. Olmeda, A. Sen,
S. K. St Angelo, Y. Cao, T. E. Mallouk, P. E. Lammert,
and V. H. Crespi, J. Am. Chem. Soc. 126, 13424 (2004).
[19] E. W. Montroll and M. F. Shlesinger, in On the Wonderful
World of Random Walks, edited by J. L. Lebowitz and
E. W. Montroll (North-Holland, Amsterdam, 1984).
[20] J. R. Howse, R. A. L. Jones, A. J. Ryan, T. Gough, R.
Vafabakhsh, and R. Golestanian, Phys. Rev. Lett. 99,
048102 (2007).
[21] N. G. Van Kampen, Stochastic Processes in Physics and
Chemistry (Elsevier, New York, 2011).
[22] J. S. Guasto, R. Rusconi, and R. Stocker, Annu. Rev. Fluid
Mech. 44, 373 (2012).
[23] E. F. Keller and L. A. Segel, J. Theor. Biol. 30, 225
(1971).
[24] W. Alt, J. Math. Biol. 9, 147 (1980).
[25] R. B. Dickinson and R. T. A Tranquillo, J. Math. Biol. 31,
563 (1993).
[26] G. Maheshwari and D. A. Lauffenburger, Microsc. Res.
Tech. 43, 358 (1998).
[27] Y. Tsori and P.-G. de Gennes, Europhys. Lett. 66, 599
(2004).
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